Abstract
In this paper, a new unified progressive hybrid censoring scheme has been constructed. This unified censoring scheme covers eleven famous censoring schemes. The estimation problem of Burr-X distribution parameters has been studied using the maximum likelihood and Bayes approaches based on the suggested unified progressive hybrid censored samples. Two real data sets have been used as illustrative engineering examples.
1. Introduction
For reasons of cost and time, life testing studies must be terminated before all failures are observed. Data censoring techniques are extensively employed to reduce test duration and costs. Both conventional type-I and type-II censoring are commonly employed in life testing. The progressive type-II censoring scheme is currently the most used. As an illustration, consider the following: assume that is a predetermined number of failures in a life-time test of n identical units or devices, where . surviving units are deleted from the test when the first failure occurs. Whenever the second failure takes place, the remaining units are also removed from the test. This goes on until the -th failure is observed, where the test is terminated and the remaining surviving units are removed. For more details, see [9, 10].
Many censoring schemes that are linked to the preceding censoring scheme are established, including type-I progressive hybrid censoring scheme [13], whenever the life testing experiments are stopped at time , where and are predetermined, type-II progressive hybrid censoring scheme [13], that stopes the experiment at time , generalized [20], that stopes the experiment at time , where , and generalized [3, 6, 7], that terminates the experiment at time , where .
The contribution of this paper is suggesting a new unified progressively hybrid censoring scheme that covers eleven famous censoring schemes. Also, two real data sets are used as illustrative examples in the engineering field.
The suggested censoring scheme can be described as follows.
For predetermined, and , where , in the event that the failure number happens before time , the experiment should be stopped at time . In case of the failure number happens at the interval between and , the experiment should be stopped at the time , and in case of the failure number happens after time , then the experiment should be stopped at time . This censoring scheme allows us to guarantee that the experiment will be completed with at most time with at least failures, and if it is not, we can guarantee precisely fails. Thus, we have the following with six cases (according to the relationship between , and ).(1)If , then the experiment should be stopped at time (2)If , then the experiment should be stopped at time (3)If , then the experiment should be stopped at time (4)If , then the experiment should be stopped at time (5)If , then the experiment should be stopped at time (6)If , then the experiment should be stopped at time
It is clear that the proposed includes a second termination time in addition to and the second number in addition to in order to provide more flexibility than the generalized type-I and generalized type-II , also to have more observations which will develop the inference.
Assume that there have been failures up to the time , . Then, given a sample related to this censoring scheme, the likelihood function will be in the following form:
Special cases:
From the suggested , many well-known censoring schemes can be obtained as follows:(1)Unified HCS [8] when (2)Generalized type-I PHCS [20] when (3)Generalized type-II PHCS [3, 6, 7] when and (4)Generalized type-I HCS [11] when and (5)Generalized type-II HCS [11] when , and (6)Type-I PHCS [13] when and (7)Type-II PHCS [13] when and (8)Type-I HCS [12, 19] when and and (9)Type-II HCS [12] when and and (10)Type-I censoring [18] when , , and (11)Type-II censoring [18] when , and and
A random variable is said to have a Burr-X with vector of parameters if its is given by
The corresponding cumulative distribution function and reliability function are given, respectively, as
Estimation and prediction problems are studied under the Burr model by many authors, see [1, 2, 17, 22].
The paper is organized as follows: in Section 2, the point and interval estimation problems have been studied using the maximum likelihood (classical) approach. In Section 3, the same problems have been studied using the Bayesian approach. The point and interval estimates of the suggested distribution parameters are obtained based on simulated and real data in Section 4. Finally, the paper is concluded in Section 5.
2. Maximum Likelihood Estimation
In this section, the (1) has been maximized to obtain the maximum likelihood estimates of the parameters and . Then, the approximate and bootstrap confidence intervals of the same parameters have been obtained.
2.1. Point Estimation
Based on the suggested , the of the parameters and given a progressively type-II censored sample , which may be written as , can be obtained by substituting (2) and (4) in (1) as follows:
To obtain the of and , denoted by and , the values of the parameters and which maximize the function are obtained numerically.
2.2. Approximate Confidence Interval
Under the assumption that is approximately bivariate normal with mean and covariance matrix , which can be approximated by , where
The approximated confidence intervals for and may be determined, respectively, by using the following formulas:where and are the main diagonal elements of the covariance matrix and is the standard normal variate such that .
2.3. Bootstrap Confidence Interval
In this section, based on the idea of [14], bootstrap confidence intervals have been obtained as follows:(1)Based on the suggested unified progressive hybrid censored sample with censoring scheme , and selected , and , the of the parameters and , and have been obtained(2)Using the estimated parameters in step 1, a progressive type-II censored sample , with the same censoring scheme , is generated(3)Based on the unified progressive hybrid censored sample related to and the same values of , and , the of the parameters and say and have been obtained(4)Repeat steps 2 and 3 times(5)After arranging all ’s and ’s in an ascending order to be and , and will be the two-sided confidence intervals for and
3. Bayes Estimation
In this section, the Bayes estimates (point and interval) have been obtained using the squared error loss function and the bivariate prior used in [4–7] which of the following form:where , and are the prior parameters.
3.1. Point Estimation
Using the squared error loss function, the of a general continuous function of the vector of parameters is given bywhich can be approximated bywhere is a sample generated from the posterior using Gibbs sampler and Metropolis-Hastings techniques, see [16, 21, 23, 24].
For the studied distribution, and the posterior , using the prior (8) and the (5), can be written in the following form:where are normalizing constants.
Using Gibbs sampler and Metropolis-Hastings techniques to generate the sample and then using formula (10), the of the parameters and will be of the following forms:and
3.2. Credibility Interval
For a certain , credibility intervals for the parameters and are two intervals and , respectively, satisfy the following conditions:where and can be obtained using the previously generated sample from as follows:
Simple formulas for computing the credibility intervals for and have been found by substituting from (15) into (14), and they are as follows:
Solving the set of nonlinear (16), the for and can be obtained (see [5, 6]).
An alternative simple technique can be used to calculate the for the parameters and . This technique can be summarized in the following steps (see [5, 6]):(1)Generate from (2)Arrange all ’s and ’s in an ascending order to be and (3)A two-sided for , say is then given by (4)Also, a two-sided for , say is given by
3.3. Highest Posterior Density Interval
To calculate the highest posterior density interval for the parameter , , we should solve the following two nonlinear equations (see [5, 6]):Similarly, the interval for , , can be calculated by solving the following two nonlinear equations (see [5, 6]):
4. Results
This section contains two subsections. In the first, the estimation problem has been studied based on simulated samples, and in the second, the same problem has been studied based on real data sets as an illustrative example.
4.1. Simulated Results
In this subsection, point and interval and estimates based on generated samples under have been calculated for different values of , and as shown in the following steps:(1)For , and , the parameters and have been generated from the (8), see [15](2)Using the generated and , a progressive type-II censored sample of size and censoring size from Burr-X has been generated, see [9](3)For different values of and , the point and interval and estimates of the parameters and have been computed(4)Repeating steps , times, the mean squared errors of all estimates have been calculated to study the behaviors of the of the parameters with respect to the values , and (5)A and interval estimates of the parameters and have been obtained for different values of , and (6)From a high number of runs, some of the obtained results have been summarized in Tables 1 and 2
From Tables 1 and 2, the following remarks can be observed:(1)The Bayes technique is better than the maximum likelihood technique because it gives smaller and narrower confidence intervals for the estimates of the parameters(2)In the maximum likelihood interval estimation problem, the confidence intervals given by the bootstrap method are narrower than that given by the approximate method(3)In the Bayesian interval estimation problem, the highest posterior confidence intervals are narrower than the credibility confidence intervals(4)In each technique of estimation, for any three fixed values from , and , the have been decreased, and the confidence intervals have been become narrower by increasing the fourth value(5)In each technique of estimation, for fixed , and , the have been increased, and the confidence intervals have become wider by increasing .
4.2. Data Analysis
In this section, two real data sets, from [25], have been introduced. In [25], the validity of Burr-XII distribution for the given real data sets is discussed, and it is found that Burr-XII distribution fits quite well for the data sets.
In this section, the same real data sets have been analyzed using the Burr-X distribution, and a comparison between the two distributions has been made using the Kolmogorov–Smirnov test statistic and it’s to select the best from the two distributions. These real data sets are as follows:
Data 1. 0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.76, 4.85, 6.5, 7.35, 8.01, 8.27, 12.06, and 31.75 which represent the failure times in hours of 15 devices.
Data 2. 0.9, 1.5, 2.3, 3.2, 3.9, 5.0, 6.2, 7.5, 8.3, 10.4, 11.1, 12.6, 15.0, 16.3, 19.3, 22.6, 24.8, 31.5, 38.1, and 53.0, which represent the first failure times in months of 20 electronic cards.
In Table 3, the maximum likelihood estimates, the corresponding Kolmogorov–Smirnov test statistic, and its have been computed under Burr-X and Burr-XII models.
Under significance level (0.05), the computed for Burr-X and Burr-XII distributions are greater than the chosen significance level which means that the two distributions fit the two real data sets well. But, the corresponding to Burr-X distribution is greater than that computed for Burr-XII which means that the Burr-X distribution fits the data better than the Burr-XII distribution.
Point and interval estimates of the parameters and based on the introduced from the given real data sets have been obtained and summarized in Table 4 using , and (for sample I) and (for sample II).
5. Conclusions
In this paper, the point and interval estimates of the Burr-X parameters have been obtained based on a suggested with complete description for the behaviors of the and the confidence intervals lengthes with respect to the values , and . From the obtained results, we concluded that the Bayes approach of estimation gives more accurate point estimates and narrower interval estimates. Moreover, in the two approaches of estimation, by increasing one of the values , and and fixing the other values, the results will be more accurate, also, by increasing with fixed , and , and we will get a lower accuracy. Two engineering real data sets have been introduced and analyzed using Burr-XII and Burr-X distribution, and we have found that the studied distribution fits the given real data sets well. Based on the two real data sets and using the suggested , the point and interval estimates of the used distribution parameters have been obtained using different techniques.
For future work, we can study the prediction problem based on the which suggested in this paper and also extend the results of this paper to other distributions. Furthermore, we can study the estimation problem under the joint .
Data Availability
All data are available in the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.